Modeling and Control in Solid Mechanics

1 Introduction.- 1 Elements of mathematical analysis and calculus of variations.- 1.1 Functional spaces. Simple properties.- 1.2 Variational inequalities.- 1.3 Minimization problems for convex functionals.- 1.4 Derivative of a convex functional.- 1.5 Minimization problems for nonsmooth functionals.- 1.6 Weak convergence. Compactness principles.- 1.7 Weak semicontinuity of functionals.- 1.8 Existence of solutions to the minimization problem.- 1.9 The case of Hilbert space.- 1.10 Elements of measure theory.- 2 Mathematical models of elastic bodies. Contact problems.- 2.1 Linear elastic bodies and shallow shells.- 2.2 Mathematical models of contact problems.- 2 Variational Inequalities in Contact Problems of Elasticity.- 1 Contact between an elastic body and a rigid body.- 1.1 Problem formulation.- 1.2 Regularity of solutions. Construction of measures.- 2 Contact between two elastic bodies.- 2.1 Formulation of the problem. Regularity of solutions.- 2.2 Construction of a measure.- 3 Contact between a shallow shell and a rigid punch.- 3.1 Existence of solutions.- 3.2 Regularity of solutions.- 3.3 Absence of concentrated forces.- 3.4 Parallel punch.- 4 Contact between two elastic plates.- 4.1 Problem formulation. Properties of the solution.- 4.2 Connectedness of the noncoincidence domain.- 5 Regularity of solutions to variational inequalities of order four.- 5.1 The contact problem of a plate with a membrane.- 5.2 The contact problem for a shell.- 6 Boundary value problems for nonlinear shells.- 6.1 General remarks.- 6.2 Inequalities on the boundary. Convergence of solutions.- 7 Boundary value problems for linear shells.- 8 Dynamic problems.- 8.1 Variational inequality for a beam.- 8.2 Variational inequality for a shell.- 3 Variational Inequalities in Plasticity.- 1 Preliminaries.- 2 The Hencky model.- 2.1 The three-dimensional elastoplastic body.- 2.2 The perfect plastic body.- 3 Dynamic problem for generalized equations of the flow model.- 4 The Kirchhoff-Love shell. Existence of solutions to the dynamic problem.- 4.1 Problem formulation.- 4.2 The main result.- 5 Existence of solutions to one-dimensional problems.- 5.1 Elastoplastic problems for a beam and cylindrical shell.- 5.2 The perfectly plastic problem for a beam.- 6 Existence of solutions for a quasistatic shell.- 6.1 Formulation of the problem.- 6.2 Theorem of existence.- 7 Contact problem for the Kirchhoff plate.- 7.1 Elastoplastic problem.- 7.2 The perfectly plastic problem.- 8 Contact problem for the Timoshenko beam.- 9 The case of tangential displacements.- 10 Beam under plasticity and creep conditions.- 11 The contact viscoelastoplastic problem for a beam.- 4 Optimal Control Problems.- 1 Optimal distribution of external forces for plates with obstacles.- 1.1 Cost functionals with measures.- 1.2 Cost functionals with norms.- 2 Optimal shape of obstacles.- 2.1 Cost functionals with norms.- 2.2 Cost functionals with measures.- 2.3 Finite set of pointwise restrictions.- 3 Other cost functionals.- 4 Plastic hinge on the boundary.- 4.1 Cost functionals with displacements.- 4.2 Cost functionals with measures.- 5 Optimal control problem for a beam.- 6 Optimal control problem for a fourth-order variational inequality.- 6.1 Fourth-order operator.- 6.2 Second-order operator.- 6.3 The passage to the limit.- 7 The case of two punches.- 7.1 Optimal control for a plate.- 7.2 Optimal control for a membrane.- 7.3 The passage to the limit.- 8 Optimal control of stretching forces.- 8.1 Optimal control for a plate.- 8.2 Optimal control for a membrane.- 8.3 Transition from a plate to a membrane.- 9 Extreme shapes of cracks in a plate.- 10 Extreme shapes of unilateral cracks.- 10.1 Interior cracks.- 10.2 Boundary cracks.- 10.3 A more precise nonpenetration condition.- 11 Optimal control in elastoplastic problems.- 12 The case of vertical and horizontal displacements.- 5 Sensitivity Analysis.- 5.1 Properties of metric projection in Hilbert spaces.- 5.2 Shape sensitivity analysis.- 5.2.1 Material derivatives.- 5.2.2 Material derivatives on the boundary ?.- 5.2.3 Shape derivatives on the boundary ?.- 5.2.4 Displacement derivatives on S.- 5.2.5 Derivatives of shape functionals.- 5.3 Unilateral problems in H20(?).- 5.3.1 The tangent cone.- 5.3.2 Differentiability of metric projections.- 5.3.3 Applications to optimal design.- 5.4 Unilateral problems in H2(?) ? H10(?).- 5.4.1 Obstacle problem for simply supported Kirchhoff plate.- 5.5 Systems with unilateral conditions.- 5.6 Shape estimation problems.- 5.6.1 Admissible domains with norm constraints on graphs.- 5.6.2 Admissible domains with local constraints on graphs.- 5.6.3 Differentiability of metric projection.- 5.6.4 Shape estimation problem for the wave equation.- 5.7 Domain optimization problem for parabolic equations.- 5.7.1 Parabolic equation in a variable domain.- 5.7.2 Differentiability of the cost functional.- 5.7.3 Shape sensitivity analysis.- 5.7.4 Optimization problem.- 5.8 Shape sensitivity analysis of thin shells.- 5.8.1 Thin shells.- 5.8.2 Displacement derivatives.- 5.8.3 Shape sensitivity analysis of thin shells.- 5.8.4 Computation of derivatives of cost functional.- 5.8.5 Computation of the second derivative.- References.